Single-molecule Forster resonance energy transfer (FRET) between fluorescent donor and acceptor labels attached to a protein or nucleic acid can be used to probe a molecules structure, dynamics and function. In these experiments, a molecule is either immobilized on a surface or diffuses through a spot illuminated by a laser, and the donor is excited. The donor can emit a photon or transfer the excitation to an acceptor which then can emit a photon of a different color. The rate of transfer depends on (interdye distance)-6 and this is why there is information about conformational dynamics (FRET is the optical analog of the NOE in NMR that is used in structure determination). The output of these experiments is a photon trajectory (the color of the photons emitted by the donor differ from those emitted by the acceptor). The observed sequence of photons can be binned, and a histogram of the FRET efficiencies for each bin, defined as the fraction of the photons emitted from the acceptor, can be constructed. The shape of the histogram depends on the conformational states of the molecule and their interconversion rates. Although this technique provides unique information that cannot be obtained at the ensemble level, the possibility of studying fast molecular dynamics is limited by the number of photons detected per unit time (photon count rate), which is proportional to the illumination intensity. To improve the dynamic range of single-molecule fluorescence spectroscopy at a given photon count rate, we consider each and every photon and use a maximum likelihood method to get the information about fast conformational dynamics. For a photon trajectory with recorded photon colors and inter-photon times, the parameters of a model describing molecular dynamics are obtained by maximizing the appropriate likelihood function. During this reporting period our investigation of the accuracy of the maximum likelihood estimates has lead to a publication(reference 1). We studied the standard deviations of the parameters of a two-state model obtained from photon sequences with recorded colors and arrival times. In the special case when the FRET efficiencies of the states are 0 and 1, an exact analytical formula was derived for the errors of the parameters extracted using by maximizing a proper likelihood function. The standard deviations can be also obtained analytically in the limiting cases of fast and slow molecule's conformational dynamics. These results are compared with the results of numerical simulations. We found that the accuracy and, therefore, the ability to predict model parameters depend on how fast the transition rates are compared to the photon count rate. In the limit of slow transitions, the key parameters that determine the accuracy are the number of transitions between the states and the number of independent photon sequences. In the fast transition limit, the accuracy is determined by the small fraction of photons that are correlated with their neighbors. The relative standard deviation of the relaxation rate has a chevron shape as a function of the transition rate in the log-log scale. The location of the minimum of this function dramatically depends on how well the FRET efficiencies of the states are separated. The above results for the standard deviations allow one to determine the amount of data required to estimate parameters to a given accuracy. In reference 2 an analytical expression is derived for the rate constant that describes diffusive transitions between two deep wells of a multidimensional potential. The expression, in contrast to the Kramers-Langer formula for the rate constant, is valid even when the diffusion is highly anisotropic. Our approach is based on a variational principle for the reactive flux and uses a trial function for the splitting probability or commitor. The theoretical result is validated by Brownian dynamics simulations. Perhaps surprisingly our result appears to be precisely what is needed to resolve a recent controversy in the field of single molecule force spectroscopy. We hope to discuss this in more detail in next year's report. Rate equations that describe the kinetics of the populations of various conformational states are widely used to understand and describe the time course of biological processes. When one tries to apply this formalism to complex reactions such a protein folding one is immediately faced with the problem of having to deal with an enormous number of states. Clearly one would like to have a model that describe adequately data ( experimental or simulated) with the fewest number of states. Therefore on is faced with the problem of how to lump or more technically aggregate states into superstates with the minimal loss of information. In reference 3, we develop a systematic procedure for obtaining rate and transition matrices that optimally describe the dynamics of such aggregated superstates. These reduced dynamical models are constructed by matching the time-dependent occupancy-number correlation functions of the superstates in the full and aggregated systems. Identical results are obtained by using a projection operator formalism. The reduced dynamic models are exact for all times in their full non-Markovian formulation. In the approximate Markovian limit, we derive simple analytic expressions for the reduced rate or Markov transition matrices that lead to exact auto- and cross-relaxation times. These reduced Markovian models strike an optimal balance between matching the dynamics at short and long times. We also discuss how this approach can be used in a hierarchical procedure of constructing optimal superstates through aggregation of microstates. The results of the general reduced-matrix theory are illustrated with applications to simple model systems and a more complex master-equation model of peptide folding derived previously from atomistic molecular dynamics simulations. We find that the reduced models faithfully capture the dynamics of the full systems, producing substantial improvements over the common local-equilibrium approximation.This approach is not only useful to construct reduced models for already defined groupings of microstates into superstates, but also helps in finding optimal superstates. Specifically, we found that maximizing the relaxation time of the reduced-matrix model provides a quantitative criterion that can be used in a hierarchical construction of superstates through aggregation of microstates.